Uniqueness of equilibrium states for some partially hyperbolic horseshoes

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January 2012, 32(1): 27-40. doi: 10.3934/dcds.2012.32.27

Uniqueness of equilibrium states for some partially hyperbolic horseshoes

Alexander Arbieto ^1, and Luciano Prudente ^1,

1.	Instituto de Matemática - UFRJ, Av. Athos da Silveira Ramos 149, Cidade Universitária - Ilha do Fundão, P.O. Box 68530. Rio de Janeiro - RJ, Brazil, Brazil

Received July 2010 Revised May 2011 Published September 2011

Abstract
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In this note, we consider a partially hyperbolic horseshoe and prove uniqueness of equilibrium states for a class of potentials. In particular we obtain that there exists a unique maximal entropy measure.

Keywords: Partially hyperbolic horseshoes, Equilibrium States, maximal entropy measure..

Mathematics Subject Classification: Primary: 37D35; Secondary: 37D3.

Citation: Alexander Arbieto, Luciano Prudente. Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 27-40. doi: 10.3934/dcds.2012.32.27

References:

[1]	J. Alves and V. Araújo, Random perturbations of non-uniformly expanding maps, Astérisque, 286 (2003), 25-62.
[2]	J. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398. doi: 10.1007/s002220000057.
[3]	A. Arbieto, C. Matheus and K. Oliveira, Equilibrium states for random non-uniformly expanding maps, Nonlinearity, 17 (2004), 581-593. doi: 10.1088/0951-7715/17/2/013.
[4]	C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. of Math., 115 (2000), 157-193. doi: 10.1007/BF02810585.
[5]	R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.
[6]	R. Bowen, Some systems with unique equilibrium states,, Math. Systems Theory, 8 (): 1974.
[7]	R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X.
[8]	R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphism," Springer Lecture Notes in Math., 470, 1975.
[9]	R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848.
[10]	H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys., 168 (1995), 571-580. doi: 10.1007/BF02101844.
[11]	H. Bruin and G. Keller, Equilibrium states for S-unimodal maps, Ergodic Theory Dynam. Systems, 18 (1998), 765-789. doi: 10.1017/S0143385798108337.
[12]	H. Bruin and M. Todd, Equilibrium states for interval maps: The potential $-t log\\|Df\\|$, Ann. Sci. École Norm. Sup., 42 (2009), 559-600.
[13]	J. Buzzi, Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states, Proc. Sympos. Pure Math., 69 (2001), 749-783.
[14]	J. Buzzi, T. Fisher, M. Sambarino and C. Vásquez, Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems,, Ergodic Theory and Dynamical Systems, ().
[15]	J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Theory Dynam. Systems, 23 (2003), 1383-1400. doi: 10.1017/S0143385703000087.
[16]	L. J. Díaz and T. Fisher, Symbolic extensions for partially hyperbolic diffeomorphisms, Discrete and Continuous Dynamical Systems, 29 (2011), 1419-1441.
[17]	L. J. Díaz, V. Horita, M. Sambarino and I. Rios, Destroying horseshoes via heterodimensional cycles: Generating bifurcations inside homoclinic classes, Ergodic Theory and Dynamical Systems, 29 (2009), 433-474. doi: 10.1017/S0143385708080346.
[18]	Haydn N.T.A. and D. Ruelle, Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification, Commun. Math. Phys., 148 (1992), 155-167. doi: 10.1007/BF02102369.
[19]	F. Hofbauer, The topological entropy of a transformation $x\mapsto ax(1-x)$, Monatsh. Math., 90 (1980), 117-141. doi: 10.1007/BF01303262.
[20]	G. Iommi and M. Todd, Natural equilibrium states for multimodal maps, Commun. Math. Phys., 300 (2010), 65-94. doi: 10.1007/s00220-010-1112-x.
[21]	R. Israel, "Convexity in the Theory of Lattice Gases," Princeton University Press, 1979.
[22]	G. Keller, Lifting measures to Markov extensions, Monatsh. Math., 108 (1989), 183-200.
[23]	F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc., 16 (1977), 568-576. doi: 10.1112/jlms/s2-16.3.568.
[24]	R. Leplaideur, K. Oliveira and I. Rios, Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes, Nonlinearity, 19 (2006), 2667-2694. doi: 10.1088/0951-7715/19/11/009.
[25]	S. E. Newhouse, Continuity properties of entropy, Annals of Mathematics, 129 (1989), 215-235. doi: 10.2307/1971492.
[26]	K. Oliveira, Equilibrium states for non-uniformly expanding maps, Ergodic Theory & Dynamical Systems, 23 (2003), 1891-1905. doi: 10.1017/S0143385703000257.
[27]	Y. Pesin and S. Senti, Equilibrium measures for maps with inducing schemes, Journal of Modern Dynamics, 2 (2008), 397-430.
[28]	V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Russian Math. Surveys, 22 (1967), 3-56.
[29]	D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Co., Reading, Mass, 5, 1978.
[30]	P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Annales de l Institut Henri Poincaré. Analyse non Linéaire, 27 (2010), 555-593.
[31]	W. Cowieson and L. S. Young, SRB measures as zero-noise limits, Ergodic Theory Dynamic Systems, 25 (2005), 1115-1138. doi: 10.1017/S0143385704000604.

show all references

References:

[1]	J. Alves and V. Araújo, Random perturbations of non-uniformly expanding maps, Astérisque, 286 (2003), 25-62.
[2]	J. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398. doi: 10.1007/s002220000057.
[3]	A. Arbieto, C. Matheus and K. Oliveira, Equilibrium states for random non-uniformly expanding maps, Nonlinearity, 17 (2004), 581-593. doi: 10.1088/0951-7715/17/2/013.
[4]	C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. of Math., 115 (2000), 157-193. doi: 10.1007/BF02810585.
[5]	R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.
[6]	R. Bowen, Some systems with unique equilibrium states,, Math. Systems Theory, 8 (): 1974.
[7]	R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X.
[8]	R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphism," Springer Lecture Notes in Math., 470, 1975.
[9]	R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848.
[10]	H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys., 168 (1995), 571-580. doi: 10.1007/BF02101844.
[11]	H. Bruin and G. Keller, Equilibrium states for S-unimodal maps, Ergodic Theory Dynam. Systems, 18 (1998), 765-789. doi: 10.1017/S0143385798108337.
[12]	H. Bruin and M. Todd, Equilibrium states for interval maps: The potential $-t log\\|Df\\|$, Ann. Sci. École Norm. Sup., 42 (2009), 559-600.
[13]	J. Buzzi, Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states, Proc. Sympos. Pure Math., 69 (2001), 749-783.
[14]	J. Buzzi, T. Fisher, M. Sambarino and C. Vásquez, Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems,, Ergodic Theory and Dynamical Systems, ().
[15]	J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Theory Dynam. Systems, 23 (2003), 1383-1400. doi: 10.1017/S0143385703000087.
[16]	L. J. Díaz and T. Fisher, Symbolic extensions for partially hyperbolic diffeomorphisms, Discrete and Continuous Dynamical Systems, 29 (2011), 1419-1441.
[17]	L. J. Díaz, V. Horita, M. Sambarino and I. Rios, Destroying horseshoes via heterodimensional cycles: Generating bifurcations inside homoclinic classes, Ergodic Theory and Dynamical Systems, 29 (2009), 433-474. doi: 10.1017/S0143385708080346.
[18]	Haydn N.T.A. and D. Ruelle, Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification, Commun. Math. Phys., 148 (1992), 155-167. doi: 10.1007/BF02102369.
[19]	F. Hofbauer, The topological entropy of a transformation $x\mapsto ax(1-x)$, Monatsh. Math., 90 (1980), 117-141. doi: 10.1007/BF01303262.
[20]	G. Iommi and M. Todd, Natural equilibrium states for multimodal maps, Commun. Math. Phys., 300 (2010), 65-94. doi: 10.1007/s00220-010-1112-x.
[21]	R. Israel, "Convexity in the Theory of Lattice Gases," Princeton University Press, 1979.
[22]	G. Keller, Lifting measures to Markov extensions, Monatsh. Math., 108 (1989), 183-200.
[23]	F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc., 16 (1977), 568-576. doi: 10.1112/jlms/s2-16.3.568.
[24]	R. Leplaideur, K. Oliveira and I. Rios, Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes, Nonlinearity, 19 (2006), 2667-2694. doi: 10.1088/0951-7715/19/11/009.
[25]	S. E. Newhouse, Continuity properties of entropy, Annals of Mathematics, 129 (1989), 215-235. doi: 10.2307/1971492.
[26]	K. Oliveira, Equilibrium states for non-uniformly expanding maps, Ergodic Theory & Dynamical Systems, 23 (2003), 1891-1905. doi: 10.1017/S0143385703000257.
[27]	Y. Pesin and S. Senti, Equilibrium measures for maps with inducing schemes, Journal of Modern Dynamics, 2 (2008), 397-430.
[28]	V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Russian Math. Surveys, 22 (1967), 3-56.
[29]	D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Co., Reading, Mass, 5, 1978.
[30]	P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Annales de l Institut Henri Poincaré. Analyse non Linéaire, 27 (2010), 555-593.
[31]	W. Cowieson and L. S. Young, SRB measures as zero-noise limits, Ergodic Theory Dynamic Systems, 25 (2005), 1115-1138. doi: 10.1017/S0143385704000604.

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